Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

June 27, 2005

Topological G2 Sigma Models

de Boer, Naqvi and Shomer have a very interesting paper, in which they claim to construct a topological version of the supersymmetric σ\sigma-model on a 7-manifold of G2G_2 holonomy. The construction is quite a bit more delicate than the usual topologically-twisted σ\sigma-model. The latter are local 2D field theories, in which the spins of the fields have been shifted in such a way that one of the (nilpotent) supercharges becomes a scalar. If you wish, you can think of them as a 3-stage process:

Start with the original “untwisted” σ\sigma-model.

Twist, to form a local, but nonunitary field theory.

Pass to the QQ-cohomology, which finally yields a unitary theory (with, in fact, a finite-dimensional Hilbert space of states).

In their construction, the observables (and, for that matter, the nilpotent “scalar” supercharge itself) are nonlocal operators, defined as projections onto particular conformal blocks in the underlying CFT. So there is no intermediate “step 2”, at least not one that is recognizable as a local field theory.

The idea that there might be a topological version of the supersymmetric σ\sigma-model on a G2G_2 manifold dates back to Shatashvili and Vafa. They noticed that, in addition to the N=1N=1 superconformal algebra (generated by T(z)T(z) and G(z)G(z)), the theory has an extended chiral algebra, with the additional generators forming supermultiplets of spin 3/2 and 2. In (1,1)superspace (with D=∂θ+θ∂zD= \partial_\theta + \theta\partial_z), we have the N=1N=1 supercurrent and stress tensor
G+θT=−12gij(X)DXi∂zXj
G+\theta T = - \frac{1}{2} g_{i j}(X) D X^i \partial_z X^j
as well as
GI+θK=i15ϕijk(3)(X)DXiDXjDXk
G_I + \theta K = \frac{i}{15} \phi^{(3)}_{i j k}(X) D X^i D X^j D X^k
and
TI+θM=15(ϕijkl(4)DXiDXjDXkDXl+12gij(X)DXi∂zDXj)
T_I + \theta M = \frac{1}{5} ( \phi^{(4)}_{i j k l} D X^i D X^j D X^k D X^l + \frac{1}{2} g_{i j}(X) D X^i \partial_z D X^j)
formed out of the covariantly-constant 3-form, ϕ(3)\phi^{(3)}, and its Hodge dual, ϕ(4)=*ϕ(3)\phi^{(4)}=*\phi^{(3)}, associated to the existence of a G2G_2 structure. The key facts are

GIG_I and TIT_I form a second superconformal algebra, with central charge c=7/10c= 7/10, i.e. there’s a hidden Tricritical Ising Model in this theory.

If we write T=TI+TrT=T_I+T_r, then TI(z)Tr(w)=T_I(z) T_r(w)= nonsingular, which is to say that, thought of as a conformal (as opposed to superconformal) theory, this model is the tensor product of a c=7/10c=7/10 Tricritical Ising Model and a second theory with c=49/5c= 49/5, whose stress tensor is TrT_r.

de Boer et al show that there’s a BPS bound on the conformal weight
h=hI+hr≥1+1+80hI8
h = h_I + h_r \geq \frac{1+\sqrt{1+80 h_I}}{8}
which is saturated for the following conformal primaries in the NS sector: |hI,hr⟩=|0,0⟩|h_I,h_r\rangle=|0,0\rangle, |1/10,2/5⟩|1/10, 2/5\rangle, |6/10,2/5⟩|6/10,2/5\rangle and |3/2,0⟩|3/2,0\rangle, whose Tricritical Ising components are just the primaries Φn,1\Phi_{n,1}, n=1,2,3,4n=1,2,3,4 in the Kač table.

The spin field, which creates the ground state of the Ramond sector, has h=7/16h=7/16, and lies entirely in the Tricritical Ising sector of the theory (it is Φ1,2\Phi_{1,2} in the Kač table). We can decompose it into two conformal blocks
Φ1,2=Φ1,2++Φ1,2−
\Phi_{1,2} = \Phi_{1,2}^+ + \Phi_{1,2}^-
defined by its action on the two Virasoro representations that comprise the R-sector (ℋ1,2\mathcal{H}_{1,2}, with hI=7/16h_I=7/16 and ℋ2,2\mathcal{H}_{2,2}, with hI=3/80h_I=3/80):
Φ1,2+:{ℋ1,2→ℋ4,1ℋ2,2→ℋ3,1Φ1,2−:{ℋ1,2→ℋ1,1ℋ2,2→ℋ2,1
\Phi_{1,2}^+ :\, \left\{ \array{
\mathcal{H}_{1,2}\to \mathcal{H}_{4,1}\\
\mathcal{H}_{2,2}\to \mathcal{H}_{3,1}}\right.\qquad
\Phi_{1,2}^- :\, \left\{ \array{
\mathcal{H}_{1,2}\to \mathcal{H}_{1,1}\\
\mathcal{H}_{2,2}\to \mathcal{H}_{2,1}}\right.
Let 𝒪n,α\mathcal{O}_{n,\alpha} be the operators corresponding to the “special” NS conformal primary states introduced above, |hI=(2n−3)(n−1)10,hr=(4−n)(n−1)5,α⟩|h_I = \frac{(2n-3)(n-1)}{10}, h_r = \frac{(4-n)(n-1)}{5},\alpha\rangle, where α\alpha is some discrete index labeling the possibly distinct operators with these conformal weights. (𝒪1=𝟙\mathcal{O}_1=&#x1D7D9; and 𝒪4=GI(z)\mathcal{O}_4=G_I(z) presumably don’t need such a label if the “internal” c=49/5c=49/5 theory is unitary.) The 𝒪n,α\mathcal{O}_{n,\alpha} don’t commute with QQ, but
𝒜n,α(z)=∑mPn+m−1𝒪n,α(z)Pm
\mathcal{A}_{n,\alpha}(z)= \sum_m P_{n+m-1} \mathcal{O}_{n,\alpha}(z) P_m
do. The observables of the “topological” theory are defined as

where I’ve suppressed the right-movers, as in the rest of my summary. The claim is that precisely these amplitudes, for n=1,…,4n=1,\dots,4, are independent of the insertion points, and constitute a 2D topological field theory.

de Boer et al also have a proposal for a higher-genus “topological string theory” generalization, but I have to say that I don’t really understand it. So, maybe I’d better stop here.

Update (6/27/2005):

I should say that the definition (1) given in their paper doesn’t make too much sense, as written. A better definition is

(2)

limw→∞w3⟨𝒜4(w)𝒜n1,α1(z1)…𝒜nk,αk(zk)⟩
\lim_{w\to\infty} w^3 \langle \mathcal{A}_{4}(w)\mathcal{A}_{n_1,\alpha_1}(z_1)\dots \mathcal{A}_{n_k,\alpha_k}(z_k)\rangle
This phrases everything in terms of NS-sector conformal blocks, and does not make the origin a distinguished point. Roughly, it corresponds to the usual notion of twisting the sphere amplitudes by putting a background charge at ∞\infty. The subtlety is that it requires, as with their whole construction, a projection onto particular conformal blocks to extract the topological amplitude.

Posted by distler at June 27, 2005 1:24 AM

TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/580